A category is a bridge between categories and if these are disjoint full subcategories of and has no more objects. In notation: .
In other words, if denotes the category of 2 objects (name them ) and 2 nonidentity arrows, which are inverses of each other, then a bridge is a category over , i.e. a category equipped with a functor . The full subcategories and are called banks of the bridge.
The arrows in the -preimage of the two arrows and in are referred to as through arrows or heteromorphisms in the bridge.
Throughout the note, letters denote objects of and letters denote objects of .
Note that if there are no heteromorphisms of the form , or if we delete them, then we receive a profunctor that we denote by . We define similarly.
Also note that a bridge is thus built up by these two profunctors and , connected by a so called Morita context which is exactly the structure that is needed to provide composition of consecutive zigzag heteromorphisms, i.e. those of the form or .
Since both banks are assumed to be full subcategories, these zigzag compositions must exist purely within the bank or .
- As mentioned, any profunctor or is in particular, a bridge, where either or empty.
- Any two full subcategories of a category determines a bridge by taking disjoint copies of and and adding heteromorphisms from .
To give a specific example, we obtain a bridge between groups and commutative semigroups with semigroup homomorphisms as heteromorphisms.
- In particular, every category has an identity bridge where every object is duplicated and every morphism is present in 4 instances. Note that this, as a category, is equivalent to itself.
Given an arbitrary profunctor , we can formally invert those heteromorphisms which are both reflection and coreflection arrows at once.
So, we can define a bridge for any
by adding heteromorphisms of the form , which would represent the formal composition , where we need to regard the formal compositions and equal whenever and . and we need to consider every consequences of these equations.
In other words, we quotient out by the smallest equivalence relation that is closed under compositions and generated by all pairs that satisfy .
Assume and are heteromorphisms, then the following compositions are defined by means of the unique arrow in [resp. in ] that exists by the coreflection [resp. reflection] property of to make the diagrams commutative:
4. Consider the category whose objects are natural numbers and arrows are matrices over a given field , composition is matrix multiplication. We can define a natural profunctor to the category of vector spaces over , by setting a through arrow to be an -tuple of elements of , i.e. .
Compositions are defined straightforwardly. A through arrow will be a reflection and coreflection arrow at once iff it is a basis of .
Applying the bridge construction, we receive a category where the object is effectively isomorphic to every dimensional vector space.
5. Let be the profunctor with heteromorphisms the element-like relations , i.e. those that satisfy
for the maximal element of the Boolean algebra
if not then
if and then
for all elements and .
We compose by defining
If is a finite set, is its power set, then the element relation is a reflection-coreflection arrow.